The use of neural networks to approximate partial differential equations (PDEs) has gained significant attention in recent years. However, the approximation of PDEs with localised phenomena, e.g., sharp gradients and singularities, remains a challenge, due to ill-defined cost functions in terms of pointwise residual sampling or poor numerical integration. In this work, we introduce $h$-adaptive finite element interpolated neural networks. The method relies on the interpolation of a neural network onto a finite element space that is gradually adapted to the solution during the training process to equidistribute a posteriori error indicator. The use of adaptive interpolation is essential in preserving the non-linear approximation capabilities of the neural networks to effectively tackle problems with localised features. The training relies on a gradient-based optimisation of a loss function based on the (dual) norm of the finite element residual of the interpolated neural network. Automatic mesh adaptation (i.e., refinement and coarsening) is performed based on a posteriori error indicators till a certain level of accuracy is reached. The proposed methodology can be applied to indefinite and nonsymmetric problems. We carry out a detailed numerical analysis of the scheme and prove several a priori error estimates, depending on the expressiveness of the neural network compared to the interpolation mesh. Our numerical experiments confirm the effectiveness of the method in capturing sharp gradients and singularities for forward PDE problems, both in 2D and 3D scenarios. We also show that the proposed preconditioning strategy (i.e., using a dual residual norm of the residual as a cost function) enhances training robustness and accelerates convergence.

翻译：近年来，利用神经网络逼近偏微分方程的方法备受关注。然而，由于基于逐点残差采样的损失函数定义不当或数值积分精度不足，对具有局部化现象（如尖锐梯度和奇异性）的偏微分方程进行逼近仍具挑战。本文提出了$h$自适应有限元插值神经网络方法。该方法通过将神经网络插值到有限元空间中，并在训练过程中逐步调整该空间以适配解的特征，从而实现后验误差指示子的均衡分布。自适应插值的运用是保留神经网络非线性逼近能力的关键，可有效处理具有局部化特征的问题。训练过程基于梯度优化，损失函数采用插值神经网络有限元残差的（对偶）范数。通过后验误差指示子自动进行网格自适应调整（即加密与粗化），直至达到指定精度。所提方法适用于不定与非对称问题。我们对该方案进行了详细的数值分析，并依据神经网络相对于插值网格的表达能力，证明了若干先验误差估计。二维与三维正问题的数值实验证实了该方法在捕捉尖锐梯度和奇异性方面的有效性。同时表明，所提出的预条件策略（即以残差对偶范数作为损失函数）可增强训练鲁棒性并加速收敛。